Simple harmonic oscillator general solution

In my mechanics textbook is given an exemple of how to find the general solution of the of the equation of motion for a force -kx (the simple harmonic oscillator problem).

He begin his analysis and finds that e^(iwt) and e^(-iwt) are both solutions. Hence C1*e^(iwt) and C2*e^(-iwt) are also solutions and therefor C1*e^(iwt) + C2*e^(-iwt) is also a solution and since it SEEMS to have 2 arbitrary constants in it, it could be the general solution. He then says that C1 and C2 MUST be complex in order for this to be the general solution.

I'm guessing he's implying that if C1 and C2 are both real, then we can show that C1*e^(iwt) + C2*e^(-iwt) turns out to have really just ONE arbitrary constant in it.

C1 and C2 should be complex numbers so as to make the solution real and to satisfy the initial conditions. If C1 and C2 are real, the solution can be real only if C4 =C1-C2=0 therefore only one arbitrary constant remains for two initial conditions.

OK, so strictly mathematically speaking, the statement that C1 and C2 are elements of R do NOT make for a general solution of the d.e. is WRONG. Any C1, C2 elements of the sedenions and beyond make up for a solution.

But BECAUSE this equation represents a physical phenomenon, we are INTERESTED (the keyword missing in my textbook) only in those C1 and C2 that make x(t) REAL. And for that to be so, not only must C1 and C2 be complex, they must also be the complex conjugate of each other.

OK, so strictly mathematically speaking, the statement that C1 and C2 are elements of R do NOT make for a general solution of the d.e. is WRONG. Any C1, C2 elements of the sedenions and beyond make up for a solution.

But BECAUSE this equation represents a physical phenomenon, we are INTERESTED (the keyword missing in my textbook) only in those C1 and C2 that make x(t) REAL. And for that to be so, not only must C1 and C2 be complex, they must also be the complex conjugate of each other.

Is that what you are saying?

You're overlooking the basic fact that the functions you have chosen for your fundamental solutions are themselves complex and by restricting the arbitrary constants to real values you have excluded almost all possible solutions of the differential equation, i.e. you have lost generality.

But BECAUSE this equation represents a physical phenomenon, we are INTERESTED (the keyword missing in my textbook) only in those C1 and C2 that make x(t) REAL. And for that to be so, not only must C1 and C2 be complex, they must also be the complex conjugate of each other.

Out of curiosity, are there C1 and C2 elements of the quaternions or some higher hyper-complex set that can make it so x(t) turn out real? Or maybe C1 elements of the complex and C2 element of the quaternions or some other mix?